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NonNon--UniformUniformCantor Ring ArraysCantor Ring Arrays
Frederick Diaz
University of Pennsylvania
Advisors:Dr. Dwight L. Jaggard
andAaron D. Jaggard
By Frederick Diaz
The Geometry of NatureThe Geometry of Nature
l The Euclidean Perspective
l The Non-Euclidean Geometry of Nature•Clouds are not spheres
• Mountains are not cones
• A coastline is not a curve
• Lightning is not composed of straight lines
• Circles, rectangles, triangles
• Spheres, cubes, cylinders, cones
By Frederick Diaz
What is a Fractal?What is a Fractal?
l Self-similarity
l Dimension
l Lacunarity
By Frederick Diaz
Cantor SetCantor Set
Stage 0
Stage 1
Stage 2
Stage 3
Stage 5
Stage 4
( )( )( )
( )( ) 63093.03ln
2ln
1ln
ln===
εεN
D
By Frederick Diaz
Antennas and ArraysAntennas and Arrays
l Single Antenna à Broad Radiation Pattern
l Array à Point-to-Point and Preferred Coverage
• Television networks
• Radio stations
• Flight towers for pinpointing location of planes
• Medical imaging and scanning
By Frederick Diaz
Array FactorArray Factor
lArray Factor• A measure of the far-field radiation of an
antenna or an antenna array
z
y
x
Problem Geometry
θ
arraytonormalθ
Array Factor:Side View
Mainbeam
Sidelobes
( )θα sinto≈
By Frederick Diaz
Traditional Array ConfigurationsTraditional Array Configurations
l Along a Periodic Lattice
l Random within a Given Area
• Low sidelobes
• Robust in terms of element failure and element
positioning
Random Circular Array
200 Elements
Periodic Lattice
200 Elements (20x20)
By Frederick Diaz
Fractal ArraysFractal Arrays
l Random Fractals => Low Sidelobes + Robustness
l Combine order and disorder to restrict randomness (tethering)
By Frederick Diaz
Creating Cantor ArraysCreating Cantor Arrays
Stage 0
Stage 1
Stage 2
Stage 3
Stage 5
Stage 4
( )( )( )
( )( ) 63093.03ln
2ln
1ln
ln===
εεN
D
3-GAP CANTOR BARSWITH DIFFERENT LACUNARITY
By Frederick Diaz
Motivation and Reference PointsMotivation and Reference Points
Base Case: Cantor Ring Array ( N = 556 )
3-gap Cantor Bar, Stage 2, D = 9/10
Jaggard and Jaggard’s Cantor Ring Arrays
FINITEWIDTH
DISCRETEINFINITESIMALWIDTH
By Frederick Diaz
GoalsGoals
l Design discrete Cantor ring arrays comparable to continuous case
l Design arrays superior to or comparable to periodic and random cases with regard to:
• Number of elements
• Sidelobe level
• Visible range
• Robustness
• Mainbeam quality
By Frederick Diaz
Azimuthal Spacing:Tethered (Range 2pi/n)
Azimuthal Spacing:Tethered (Range (3/4)2pi/n)Taper Out Inverse Taper
l Periodic
l Random
l Tethered
l Taper Out
l Inverse Taper
l Fractal
Azimuthal Azimuthal Element Element ConfigurationsConfigurations
Azimuthal Spacing: Periodic Azimuthal Spacing: Random
By Frederick Diaz
Azimuthal Spacing:PeriodicN = 556
Azimuthal Spacing:Tethered (Range (3/4)2pi/n)
N = 556
-3 dB
l Azimuthal Spacing: Periodic à Tetheredl Lowered sidelobe power by factor of 2 when number of
elements kept constant
l Achieved comparable performance using half as
many elements
Project ResultsProject Results
By Frederick Diaz
Next StepsNext Steps
l Other Element Configurations• Tapered
• Inverse-tapered
• Fractal in azimuthal direction
